In the case when \(\Phi = L^p\)  with \(p\in [1,\infty ]\)  we get the classical vector-valued Besov and Triebel-Lizorkin spaces. For a coherent treatment in this vector setting see, e.g. [1]}, where historical remarks are included (see also Triebel [2]} for the scalar case). Recall that, if \(X\)  is a Hilbert space, then \(F^{m,2}_{L^p}(X) = W^{m, p}(X)\)  for \(m\in \mathbb {N}\)  and \(F^{s,2}_{L^p}(X) = H^{s, p}(X)\) , \(s\in \mathbb {R}\) . The case when \(\Phi = L^p_w\)  with \(w\)  in the Muckenhoupt class \(A_\infty \)  was recently considered, e.g. in [3]}, [4]}; see also the references given therein. Here and in the sequel, by \(L^p_w\)  we denote the weighted Lebesgue space \(L^p(\mathbb {R}, w\mathrm {d}t; \mathbb {C})\) .
